Problem: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{q^2 - 49}{q - 7}$
Answer: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = q$ $ b = \sqrt{49} = -7$ So we can rewrite the expression as: $t = \dfrac{({q} {-7})({q} + {7})} {q - 7} $ We can divide the numerator and denominator by $(q - 7)$ on condition that $q \neq 7$ Therefore $t = q + 7; q \neq 7$